We will describe the topologies on the visual compactification ¯X “ X Y B8X and on the
τmod-bordification ¯X τmod
“ X Y BτmodX in terms of shadows and related “basic subsets”.
We need the following notions of shadows at infinity in B8X and BτmodX.
Definition 3.71 (Shadows at infinity). (i) For points x, y P X we define the shadow of the point y as seen from x by
pShx,y :“ tξ : y P xξu Ă B8X,
and for r ą 0 the shadow of the open r-ball around y by
bShx,y,r:“ tξ : xξ X Bpy, rq ‰ Hu Ă B8X.
(ii) For points x, y P X we define the τmod-shadow of the point y as seen from x by
pShτmod
x,y :“ tτ : y P V px, stpτ qqu Ă BτmodX,
and for r ą 0 the τmod-shadow of the open r-ball around y by
bShτmod
x,y,r :“ tτ : V px, stpτ qq X Bpy, rq ‰ Hu Ă BτmodX,
By coning off the shadows at infinity at points in X and removing large balls around their tips, we obtain the subsets of ¯X and ¯Xτmod
which we will use to describe, respectively, construct the natural topologies.
Definition 3.72 (Basic subsets). (i) For points x, y P X and radii r ą 0, we define the subsets
pOx,y :“ tz : z ‰ y P xzu Ă X,
and
bOx,y,r:“ tz : xz X Bpy, rq ‰ Hu Ă X,
and the basic subsets
p ¯Ox,y :“ pOx,yY pShx,y Ă ¯X and b ¯Ox,y,r :“ bOx,y,rY bShx,y,rĂ ¯X.
(ii) For points x, y P X and radii r ą 0, we define the subsets pOτmod
x,y :“ tz : xz τmod-regular and z‰ y P ♦τmodpx, zqu Ă X
and
bOτmod
x,y,r :“ tz : xz τmod-regular and ♦τmodpx, zq X Bpy, rq ‰ Hu Ă X,
and the τmod-basic subsets
p ¯Oτmod x,y :“ pO τmod x,y Y pSh τmod x,y Ă ¯X τmod and b ¯Oτmod
x,y,r :“ bOx,y,rY bShx,y,r Ă ¯X τmod
We observe the following relations between point and ball shadows: bShx,y,r“ ď zPBpy,rq pShx,z and bShτmod x,y,r “ ď zPBpy,rq pShτmod x,z
There are analogous relations between point and ball type basic subsets: bOx,y,r“ ď zPBpy,rq pOx,z and bO τmod x,y,r “ ď zPBpy,rq pOτmod x,z
We note that the τmod-versions of the shadows and basic subsets are generalizations of these to
arbitrary rank and agree with them in rank one.
We first recall the description of the visual topology on the visual compactification ¯X. Fact 3.73. (i) For every point x P X, the basic subsets bOx,¨,¨ form together with the open
subsets of X a basis of the visual topology on ¯X.
(ii) For every ray xξ Ă X, every sequence yn Ñ 8 of points yn P xξ and every bounded
sequence of radii rną 0 the basic subsets bOx,yn,rn form a neighborhood basis of ξ.
This restricts to the following description of the visual topology on B8X.
Fact 3.74. (i) For every point x P X, the shadows bShx,¨,¨ form a basis of the visual topology
on B8X. If X is a euclidean building, then also the shadows pShx,¨ form a basis.
(ii) For every ray xξ Ă X, every sequence yn Ñ 8 of points yn P xξ and every bounded
sequence of radii rn ą 0, the shadows bShx,yn,rn form a neighborhood basis of ξ. If X is a
euclidean building, then also the shadows pShx,yn form a neighborhood basis. Moreover, if X is a symmetric space, then for x‰ y P xξ also the shadows bShx,y,¨ form a neighborhood basis.
Now we construct natural topologies onBτmodX and, at least partially, on ¯X
τmod
. Lemma 3.75. The subsets bOτmod
¨,¨,¨ are open in X. If X is a euclidean building, then also the
subsets pOτmod
¨,¨ are open.
Proof. The openness of bOτmod
¨,¨,¨ follows from the semicontinuity of diamonds, cf. Lemma 3.37.
If X is a euclidean building, then the openness of pOτmod
¨,¨ is a consequence of Corollary 3.53.
Lemma 3.76. If xy is τmod-regular, then y P Bpy, rq Ă bOτx,y,rmod for all sufficiently small rą 0.
Proof. If r is sufficiently small, then the segments xz are τmod-regular for all zP Bpy, rq.
Lemma 3.77. (i) If z P bOτmod
x,y,r, then there exists są 0 such that b ¯O τmod
x,z,s Ă b ¯O τmod
x,y,r.
(ii) If X is a euclidean building and zP pOτmod
x,y , then there is s ą 0 with b ¯O τmod
x,z,s Ă p ¯O τmod
x,y .
Proof. (i) Due to the semicontinuity of diamonds, see Lemma 3.37, there exists s ą 0 such that for every z1 P Bpz, sq the segment xz1 is τ
mod-regular and the diamond ♦τmodpx, z
1q still
(ii) The argument is the same, but using Corollary 3.53. It implies that there exists są 0 such that for every z1 P Bpz, sq the segment xz1 is τ
mod-regular and the diamond ♦τmodpx, z
1q
still contains y.
Corollary 3.78. (i) The subsets b ¯Oτmod
x,¨,¨ form together with the open subsets of X the basis of
a topology Tx on ¯X τmod
. If X is a euclidean building, then also the subsets p ¯Oτmod
x,¨ form a basis.
(ii) For every simplex τ P BτmodX, every asymptotically uniformly τmod-regular sequence
yn Ñ 8 in V px, stpτ qq and every bounded sequence of radii rn ą 0, the basic subsets b ¯O τmod
x,yn,rn
form a neighborhood basis for τ inp ¯Xτmod
, Txq. If X is a euclidean building, then also the subsets
p ¯Oτmod
x,yn form a neighborhood basis. In particular, Tx is first-countable.
Proof. (i) Suppose that τ belongs to a finite intersectionXibShτx,ymodi,ri. This means that Vpx, stpτ qq
intersects all balls Bpyi, riq. Let z P V px, ostpτ qq ´ txu be a point so that ♦τmodpx, zq also in-
tersects them. Then z P XibOτx,ymodi,ri. With the lemma it follows that τ P b ¯O
τmod
x,z,s Ă Xib ¯O τmod
x,yi,ri
for all sufficiently small s. Furthermore, XibOτx,ymodi,ri is open in X.
The subsets b ¯Oτmod
x,¨,¨ are unions of subsets of the form p ¯O τmod
x,¨ . If X is a euclidean building,
then vice versa the subsets p ¯Oτmod
x,¨ are unions of subsets of the form b ¯O τmod
x,¨,¨ by the last lemma.
(ii) Suppose that τ P bShx,y,r and that b ¯O τmod
x,yn,rn Ć b ¯O
τmod
x,y,r for all n. Then there exist points
zn P Bpyn, rnq such that xzn is τmod-regular and ♦τmodpx, znq X Bpy, rq “ H.
If X is locally compact, then after passing to a subsequence, xzn subconverges to a ray
xζ Ă V px, stpτ qq with ζ P ostpτ q. Let w P xζ be a point such that y P ♦τmodpx, wq, and let
wnP xznbe points converging to it, wnÑ w. Then ♦τmodpx, wnq X Bpy, rq ‰ H for large n, due
to the semicontinuity of diamonds, see Lemma 3.37, and hence also ♦τmodpx, znq X Bpy, rq ‰ H,
a contradiction.
If X is a euclidean building, then it follows with Corollary 3.53 that y P ♦τmodpx, znq, for
large n, which is also a contradiction. Thus, the subsets b ¯Oτmod
x,yn,rn form a neighborhood basis. If X is a euclidean building, it
follows that also the smaller open subsets p ¯Oτmod
x,yn Ă b ¯O
τmod
x,yn,rn form a neighborhood basis.
We compare now the topologies Tx for different base points x.
By construction, they all restrict to the given topology on X.
Regarding the comparison of the topologies Tx at infinity on BτmodX and on the entire
bordification ¯Xτmod
, we use that if a topological space is first-countable, then its topology is determined by the sequential convergence. Namely, a subset is a neighborhood of a point, if and only if it cannot be avoided by a sequence converging to this point. We therefore compare sequential convergence for the topologies Tx. We will do this only partially, namely for arbitrary
sequences in BτmodX, but only for asymptotically uniformly τmod-regular sequences in X. This
will be sufficient for the purposes of this paper.
We first observe that Tx-convergence translates into Hausdorff convergence of diamonds and
Weyl cones.
Hausdorff convergence Vpx, stpτnqqq X ¯Bpx, Rq Ñ V px, stpτ qq X ¯Bpx, Rq of truncated Weyl cones
for all radii Rą 0.
(ii) For an asymptotically uniformly τmod-regular sequence xn Ñ 8 in X, the convergence
xn Ñ τ in p ¯X τmod
, Txq is equivalent to the Hausdorff convergence ♦τmodpx, xnq X ¯Bpx, Rq Ñ
Vpx, stpτ qq X ¯Bpx, Rq of truncated diamonds for all radii R ą 0
Proof. The first statement follows from the second one in view of Lemma 3.35.
For the second statement, suppose that xn Ñ τ . Then for every point y P V px, stpτ qq and
radius r ą 0, the diamonds ♦τmodpx, xnq intersect Bpy, rq for all sufficiently large n. Hence,
dpy, ♦τmodpx, xnqq Ñ 0 as n Ñ `8, and the continuity of diamonds (Proposition 3.70) implies
that ♦τmodpx, yq Ă Nǫnp♦τmodpx, xnqq with a sequence ǫn Ñ 0. Again in view of Lemma 3.35,
this yields the asserted Hausdorff convergence. The converse direction is clear. Corollary 3.80. The topology Tx on ¯X
τmod
is Hausdorff.
Proof. This is a direct consequence of first-countability and the last lemma, because it implies that limits of sequences are unique.
We now compare the topologies Tx onBτmodX. We do this by comparing them to the visual
topology on B8X. For every type ¯ξ P intpτmodq, there is the natural identification
θ´1p¯ξqÝÑ B1:1 τmodX (3.81)
with the subspace θ´1p¯ξq Ă B
8X, assigning to a point ξ P B8X with type θpξq “ ¯ξ the type
τmod simplex τ spanned by it, ξ P intpτ q.
Lemma 3.82. For every type ¯ξ P intpτmodq and every point x P X, the bijection (3.81) is a
homeomorphism with respect to the restrictions of the visual topology on ¯X to θ´1p¯ξq and the
topology Tx on ¯X τmod
to BτmodX.
Proof. Let pξnq and ξ be a sequence and a point in θ´1p ¯ξq Ă B8X, and let pτnq and τ be the
corresponding sequence and point in BτmodX, i.e. ξn P intpτnq and ξ P intpτ q. We must show
that ξnÑ ξ if and only if τnÑ τ with respect to the topologies in consideration.
Suppose that τn Ñ τ with respect to Tx. Then Vpx, stpτnqq Ñ V px, stpτ qq by the pre-
vious lemma. In particular, increasingly long subsegments xyn Ă xξn Ă V px, stpτnqq be-
come arbitrarily close to segments x¯yn Ă V px, stpτ qq. We want to find Hausdorff close seg-
ments in Vpx, stpτ qq of the same type ¯ξ. By the triangle inequality for ∆-lengths (2.15), }d∆px, ynq ´ d∆px, ¯ynq} ď dpyn,y¯nq Ñ 0 and, in a euclidean Weyl chamber through ¯yn with tip
x, we find a point zn P V px, stpτ qq with d∆px, znq “ d∆px, ynq. Then dpzn,y¯nq “ }d∆pzn,y¯nq} “
}d∆px, znq ´ d∆px, ¯ynq} Ñ 0, and hence dpzn, ynq Ñ 0 by the triangle inequality. Moreover,
θpxznq “ θpxynq “ ¯ξ and therefore zn P xξ, because ξ is the only point in stpτ q with type ¯ξ. It
follows that xξnÑ xξ, i.e. ξnÑ ξ.
Conversely, suppose that ξnÑ ξ, i.e. xξn Ñ xξ. Then any ball centered at xξ is intersected
Corollary 3.83. The restriction of the topology Tx to BτmodX does not depend on x.
Definition 3.84 (Visual topology). We call this topology on BτmodX the visual topology.
Now we show that the topologies Tx agree on the entire bordification ¯X τmod
“in τmod-regular
directions”. We reformulate the condition for Tx-convergence for asymptotically uniformly τmod-
regular sequences xnÑ 8 given in Lemma 3.79 above, in order to show its independence of x.
We do this separately in the symmetric space (locally compact) and euclidean building cases. In the locally compact case, we can express Tx-convergence in terms of accumulation at
infinity (the limit set) in ¯X:
Lemma 3.85. Suppose that X is locally compact. Then xn Ñ τ P BτmodX with respect to Tx,
if and only if the accumulation set of pxnq in ¯X (with respect to the visual topology of ¯X) is
contained in ostpτ q Ă B8X.
Proof. Since X is locally compact, the sequence pxnq subconverges in both ¯X and ¯X τmod
. The latter holds, because the sequence of diamonds ♦τmodpx, xnq Hausdorff subconverges and, in
view of Lemma 3.35, the Hausdorff sublimits must be type τmod Weyl cones. Note also that
pxnq accumulates in ¯X only at the τmod-regular part θ´1postpτmodqq of B8X, as a consequence
of asymptotic uniform τmod-regularity.
Therefore, if the assertion is wrong, we may assume after passing to a subsequence, that xn Ñ τ in ¯X
τmod
and xn Ñ ξ1 P ostpτ1q in ¯X for different simplices τ, τ1 P BτmodX. But then
♦τmodpx, xnq Ñ V px, stpτ qq according to Lemma 3.79. Since xxnÑ xξ
1, it follows that ξ1 P stpτ q,
a contradiction.
In the euclidean building case, we can strengthen the condition of Hausdorff convergence of Weyl cones to initial coincidence up to increasing radii.
Lemma 3.86. Suppose that X is a euclidean building. Then xn Ñ τ P BτmodX with respect to
Tx, if and only if for every R ą 0 it holds that ♦τmodpx, xnq X ¯Bpx, Rq “ V px, stpτ qq X ¯Bpx, Rq
for all sufficiently large n.
Proof. This is a consequence of Lemmas 3.79 and 3.55.
Corollary 3.87. Whether an asymptotically uniformly τmod-regular sequence xn Ñ 8 in X
converges to a simplex τ P BτmodX in p ¯X
τmod
, Txq, does not depend on x.
Proof. If X is locally compact, this follows immediately from Lemma 3.85. Assume therefore that X is a euclidean building.
Let x, x1 P X and suppose that x
n Ñ τ P BτmodX with respect to Tx. By Lemma 3.86,
there exists a sequence yn Ñ 8 of points yn P xxn X V px, stpτ qq. Let yn1 P x1xn be points
uniformly close to the points yn, e.g. such that dpy1n, ynq ď dpx1, xq. Then the sequence pyn1q
is contained in a tubular neighborhood (of radius dpx1, xq) of V px, stpτ qq, and hence also in a
Hausdorff distance (ď dpx1, xq). The sequences py
nq and pyn1q inherit from pxnq asymptotically
uniform τmod-regularity.
Consider the subsegments x1z1
n “ x1yn1 X V px1,stpτ qq. According to Corollary 3.53, the
distances dpz1
n, yn1q are uniformly bounded, and therefore z1n Ñ 8. Since ♦τmodpx
1, z1 nq Ă
♦τmodpx
1, x
nqXV px1,stpτ qq, it follows, using again Lemma 3.35, that ♦τmodpx
1, x
nq Ñ V px1,stpτ qq.
Hence, xn Ñ τ also with respect to Tx1.
The corollary justifies the following definition.
Definition 3.88 (Flag convergence). We say that an asymptotically uniformly τmod-regular
sequence xn Ñ 8 in X flag converges to a simplex τ P BτmodX, if xn Ñ τ in p ¯X
τmod
, Txq for
some base point x.
Now we can also make precise the coincidence of the topologies Tx “in τmod-regular direc-
tions”. Suppose that A Ă X is an asymptotically uniformly τmod-regular subset, and consider
the subset
˜
Aτmod :“ A Y BτmodX Ă ¯X
τmod
. Corollary 3.89. The topology induced by Tx on ˜A
τmod
does not depend on x.
Definition 3.90 (Topology of flag convergence). We call this topology on ˜Aτmod
the topol- ogy of flag convergence.
As shown above, the topologies Tx and hence the topology of flag convergence on ˜A τmod
are Hausdorff and first-countable. Neighborhood bases at infinity have been described in Corol- lary 3.78.
We further discuss the flag convergence of sequences.
A situation when an asymptotically uniformly regular sequence flag converges, is when it stays close to a Weyl cone:
Lemma 3.91. Suppose that the asymptotically uniformly τmod-regular sequence xn Ñ 8 is
contained in the tubular neighborhood of the type τmod Weyl cone Vpx, stpτ qq. Then xn Ñ τ .
Proof. If X is locally compact, this follows from Lemma 3.85.
Suppose therefore that X is a euclidean building. Consider the points ynwhere the segments
xxn exit the Weyl cone Vpx, stpτ qq, i.e. xyn“ xxnX V px, stpτ qq. Then Corollary 3.53 implies
that dpyn, xnq is bounded. Hence ynÑ 8 is an asymptotically uniformly τmod-regular sequence
in Vpx, stpτ qq, and xn P p ¯O τmod
x,yn. The basic subsets p ¯O
τmod
x,yn form a neighborhood basis of τ .
Thus, xn Ñ τ also in this case.
We give a name to this stronger form of flag convergence:
Definition 3.92 (Conical convergence, cf. [KLP2, Def. 6.1]). We say that an asymp- totically uniformly τmod-regular sequence xn Ñ 8 in X flag converges conically to τ P BτmodX
Corollary 3.93. Let Vpx, stpτ qq and V px1,stpτ1qq be type τ
mod Weyl cones. Suppose that for
some D ą 0 the intersection of their D-neighborhoods contains an asymptotically uniformly τmod-regular sequence. Then τ “ τ1.
Proof. If pxnq is such an asymptotically uniformly τmod-regular sequence, then xn Ñ τ and
xn Ñ τ1. The assertion follows from the Hausdorff property of the topologies Tx.
The following convergence criterion will be useful when X is not locally compact.
Lemma 3.94. Let xnÑ 8 be an asymptotically uniformly τmod-regular sequence in X, and let
pτnq be a sequence in BτmodX such that for some point xP X and some constant D ě 0 it holds
that xnP ¯NDpV px, stpτmqqq for all m ě n. Then the sequence pτnq converges, τnÑ τ P BτmodX,
and xn P ¯NDpV px, stpτ qqq for all n. In particular, xnÑ τ conically.
Proof. If X is locally compact, then there exists a convergent subsequence of simplices, τnk Ñ τ .
It follows that xnP ¯NDpV px, stpτ qqq for all n, and the assertion holds in this case.
Suppose therefore that X is a euclidean building (because otherwise X is locally compact). For suitable Θ, the segments xxn are Θ-regular for large n. Let τn1 P pSh
τmod
x,xn. Applying
Lemma 3.55, we obtain for any radius rą 0 that
Vpx, stpτn1qq X ¯Bpx, rq “ V px, stpτmqq X ¯Bpx, rq
for mě n ě n0prq. Thus, both sides are independent of m and n, and isometric to
Vpx, stpτmqq X ¯Bpx, rq “ Cpx, rq
for mě n0prq. The union of the nested family of cones Cpx, rq as r Ñ `8 is a type τmod Weyl
cone Vpx, stpτ qq. It follows that τm Ñ τ and xnP ¯NDpV px, stpτ qqq.